1.3: Sustainability and energy balance on Earth
- Page ID
- 121079
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- Justify the given mathematical model that describes the energy input and output on Planet Earth.
- Use the given model to determine the energy equilibrium of the planet.
The sustainability of life on Planet Earth depends on a fine balance between the temperature of its oceans and land masses and the ability of life forms to tolerate climate change. As a follow-up to our model for nutrient balance, we introduce a simple energy balance model to track incoming and outgoing energy and determine a rough estimate for the Earth’s temperature. We use the following basic assumptions:
- Energy input from the sun, given the Earth’s radius \(r\), can be approximated as
\[E_{\text {in }}=(1-a) S \pi r^{2}, \]
where \(S\) is incoming radiation energy per unit area (also called the solar constant) and \(0 \leq a \leq 1\) is the fraction of that energy reflected; \(a\) is also called the albedo, and depends on cloud cover, and other planet characteristics (such as percent forest, snow, desert, and ocean).
- Energy lost from Earth due to radiation into space depends on the current temperature of the Earth \(T\), and is approximated as
\[E_{\text {out }}=4 \pi r^{2} \varepsilon \sigma T^{4}, \]
where \(\varepsilon\) is the emissivity of the Earth’s atmosphere, which represents the Earth’s tendency to emit radiation energy. This constant depends on cloud cover, water vapour, as well as on greenhouse gas concentration in the atmosphere; \(\sigma\) is a physical constant (the Stephan-Boltzmann constant) which is fixed for the purpose of our discussion.
- Do you think \(E_{i n}\) is proportional to Earth’s surface area or volume?
Notice there are several different symbols in Equations (\(\PageIndex{1}\)) and (\(\PageIndex{2}\)). Being clear about which are constants and which are variables is critical to using any mathematical model. As the next example points out, sometimes you have a choice to make.
Explain in what sense the two forms of energy above can be viewed as power functions, and what types of power functions they represent.
Solution
Both \(E_{\text {in }}\) and \(E_{\text {out }}\) depend on Earth’s radius as the power \(\sim r^{2}\). However, since this radius is a constant, it is not fruitful to consider it as an interesting variable for this problem (unlike the cell size example in Section 1.2). However, we note that \(E_{\text {out }}\) depends on temperatureas \(\sim T^{4}\). (We might also select the albedo as a variable and in that case, we note that \(E_{\text {in }}\) depends linearly on the albedo \(a\).)
Explain how the assumptions above can be used to determine the equilibrium temperature of the Earth, that is, the temperature at which the incoming and outgoing radiation energies are balanced.
Solution
The Earth is at equilibrium when
\[E_{\text {in }}=E_{\text {out }} \quad \Rightarrow \quad(1-a) S \pi r^{2}=4 \pi r^{2} \varepsilon \sigma T^{4} \nonumber \]
We observe that the factors \(\pi r^{2}\) cancel, and we obtain an equation that can be solved for the temperature \(T\). It is instructive to examine how this temperature depends on the constants in the problem, and how it is affected by cloud cover and greenhouse gas level. This is also explored in Exercise 21